https://file.aiquickdraw.com/mathbot/file/1726071909339a3o4tfwo.jpg

The question provided involves a two-class problem, and the goal is to compute the direction of projection \( W \) using **Fisher's Linear Discriminant Analysis (LDA)**. To solve this, let's break down the steps:

1. **Class Data**:
   - Class \(\omega_1\): 
     - Points: \( (1, 2)^T \), \( (2, 3)^T \), \( (3, 3)^T \)
   - Class \(\omega_2\): 
     - Points: \( (4, 5)^T \), \( (5, 6)^T \), \( (6, 5)^T \)

2. **Step-by-step procedure**:
   To find the projection vector \( W \) using Fisher's LDA, we follow these steps:

   ### Step 1: Compute the mean vectors for both classes
   - Mean vector for class \( \omega_1 \):
     \[
     \mu_1 = \frac{1}{3} [(1, 2)^T + (2, 3)^T + (3, 3)^T] = (2, 2.67)^T
     \]
   - Mean vector for class \( \omega_2 \):
     \[
     \mu_2 = \frac{1}{3} [(4, 5)^T + (5, 6)^T + (6, 5)^T] = (5, 5.33)^T
     \]

   ### Step 2: Compute the scatter matrices
   - **Within-class scatter** matrix \( S_W \):
     \[
     S_W = S_1 + S_2
     \]
     where \( S_1 \) and \( S_2 \) are the scatter matrices for classes \( \omega_1 \) and \( \omega_2 \).

     - For class \( \omega_1 \), deviations from the mean are:
       \[
       (1, 2)^T - (2, 2.67)^T = (-1, -0.67)^T
       \]
       \[
       (2, 3)^T - (2, 2.67)^T = (0, 0.33)^T
       \]
       \[
       (3, 3)^T - (2, 2.67)^T = (1, 0.33)^T
       \]
       The scatter matrix \( S_1 \) is the sum of the outer products of these deviations:
       \[
       S_1 = (-1, -0.67)^T (-1, -0.67) + (0, 0.33)^T (0, 0.33) + (1, 0.33)^T (1, 0.33)
       \]

     - For class \( \omega_2 \), deviations from the mean are:
       \[
       (4, 5)^T - (5, 5.33)^T = (-1, -0.33)^T
       \]
       \[
       (5, 6)^T - (5, 5.33)^T = (0, 0.67)^T
       \]
       \[
       (6, 5)^T - (5, 5.33)^T = (1, -0.33)^T
       \]
       The scatter matrix \( S_2 \) is the sum of the outer products of these deviations:
       \[
       S_2 = (-1, -0.33)^T (-1, -0.33) + (0, 0.67)^T (0, 0.67) + (1, -0.33)^T (1, -0.33)
       \]

   ### Step 3: Compute the projection vector \( W \)
   The projection vector \( W \) is given by:
   \[
   W = S_W^{-1} (\mu_1 - \mu_2)
   \]
   where \( S_W^{-1} \) is the inverse of the within-class scatter matrix, and \( \mu_1 - \mu_2 \) is the difference in the mean vectors of the two classes.

### Let's compute this in Python to find the closest answer.The projection vector \( W \), calculated using Fisher's linear discriminant analysis, is approximately:

\[
W = \begin{pmatrix} -0.309 \\ -1.763 \end{pmatrix}
\]

Looking at the given options, the closest match is **Option b**:

\[
\text{b)} \begin{pmatrix} -0.308 \\ -1.769 \end{pmatrix}
\]

Would you like more details or have any further questions?

Here are 5 related questions to expand on the topic:
1. How does Fisher's Linear Discriminant work in higher dimensions?
2. Why is the within-class scatter matrix important in LDA?
3. How is the between-class scatter matrix used in more advanced LDA models?
4. What are the main differences between LDA and PCA (Principal Component Analysis)?
5. How does Fisher's LDA handle overlapping classes in datasets?

**Tip**: When applying Fisher's Linear Discriminant, ensuring good separation between class means is key for maximizing class distinction along the projection.

For a two-class problem, (1, 2), (2, 3), (3, 3) ∈ ω1 and (4, 5), (5, 6), (6, 5) ∈ ω2, find the direction of projection W using Fisher's linear discriminator.

Learn how to solve a two-class problem using Fisher's Linear Discriminant Analysis (LDA) with step-by-step computation. The example includes calculating class means, scatter matrices, and the projection vector for class distinction.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation